Electron correlation in two-photon double ionization of helium from attosecond to FEL pulses J. Feist1, R. Pazourek1, S. Nagele1, E. Persson1, B. I. Schneider2,3, L. A. Collins4, and J. Burgd¨orfer1 1 InstituteforTheoreticalPhysics,ViennaUniversityofTechnology,1040Vienna,Austria,EU 2 PhysicsDivision,NationalScienceFoundation,Arlington,Virginia22230,USA 3 ElectronandAtomicPhysicsDivision,NationalInstituteofStandardsandTechnology,Gaithersburg,Maryland20899,USA 4 TheoreticalDivision,T-4,LosAlamosNationalLaboratory,LosAlamos,NewMexico87545,USA E-mail: [email protected] 9 Abstract. We investigate the role of electron correlation in the two-photon double 0 ionization of helium for ultrashort XUV pulses with durations ranging from a 0 2 hundred attoseconds to a few femtoseconds. We perform time-dependent ab initio calculations for pulses with mean frequencies in the so-called “sequential” regime n a ((cid:126)ω >54.4eV). Electron correlation induced by the time correlation between J emission events manifests itself in the angular distribution of the ejected electrons, 6 which strongly depends on the energy sharing between them. We show that for 2 ultrashort pulses two-photon double ionization probabilities scale non-uniformly with pulse duration depending on the energy sharing between the electrons. Most ] h interestingly we find evidence for an interference between direct (“nonsequential”) p and indirect (“sequential”) double photo-ionization with intermediate shake-up - states, the strength of which is controlled by the pulse duration. This observation m may provide a route toward measuring the pulse duration of FEL pulses. o t a PACSnumbers: 32.80.Rm,32.80.Fb,42.50.Hz . s c 1. Introduction priori necessary [11, 35–40]. i s y In a previous paper [11], we investigated the role h The role of electron correlation is of central interest of energy and angular correlations in the shortest p in our understanding of atoms, molecules and solids. pulses available today, where the distinction between [ The recent progress in the development of ultrashort “sequential” and “nonsequential” becomes obsolete. In 1 and intense light sources [1–10] provides unprecedented this contribution, we explore the dependence of two- v opportunities to study the effects of correlation not photon double ionization (TPDI) on the pulse duration 2 only in stationary states, but also in transient states T ranging from 100 attoseconds (the duration of the p 7 (i.e.,resonances),andeventoactivelyinducedynamical shortest pulses p∼roduced by high-harmonic generation 0 4 correlations [11]. [3]) to a few femtoseconds (the expected duration of 1. The helium atom is the simplest atomic system FEL pulse “bursts”). Tp can be used as a control knob whereelectron-electroninteractionscanbestudied,with to change from a “direct” to an “indirect” process. In 0 9 its double ionization being the prototype reaction for a section 3, we discuss the behavior of the one-electron 0 three-body Coulomb breakup. While computationally ionization rate PDI(E)/T , which displays non-uniform p : challenging, the full dynamics of the helium atom can scaling with T . In section 4, we investigate the angular v p i still be accurately simulated in ab initio calculations correlations, with a focus on longer pulses, which reveal X [12]. With the advent of intense XUV pulses, the focus thedetaileddynamicsoftheTPDIprocess. Insection5, r has shifted from single-photon double ionization [13– weshowthatforenergiesabovethethresholdassociated a 17] and intense-IR laser ionization by rescattering ([18– with shake-up ionization of the He atom, interferences 20] and references therein) to multiphoton ionization. between sequential and nonsequential contributions can Two-photon double ionization (TPDI) has recently be observed, the strength of which can be varied by received considerable attention, both in the so-called changing the pulse duration. One consequence is that “nonsequential” or “direct” regime (39.5eV < (cid:126)ω < from the size and shape of these Fano-like resonances, 54.4eV), where the electrons necessarily have to share thepulsedurationofXUVpulsesmightbededuced. All energyviaelectron-electroninteractiontoachievedouble this information is encoded in the final joint momentum ionization[21–34], andinthe“sequential”regime((cid:126)ω > distribution PDI(k ,k ) PDI(E ,E ,Ω ,Ω ), which 1 2 1 2 1 2 ≡ 54.4eV), where electron-electron interaction is not a is experimentally accessible in kinematically complete 2 COLTRIMSmeasurements[41]. Inthiscontribution,we r = 800a.u. for the longest pulses. All presented max focus on integrated quantities, which are more readily quantities were tested for numerical convergence and accessible because of better statistics. Unless otherwise gauge independence. stated, atomic units are used. 3. Pulse length dependence of TPDI 2. Method The nature of the two-photon double ionization (TPDI) Our theoretical approach (described in more detail in process depends strongly on the photon energy. In [23]) is based on a direct solution of the time-dependent order to doubly ionize the helium atom (ground state Schr¨odinger equation (TDSE) by the time-dependent energy E0 79eV), each photon must have an close-coupling (TDCC) scheme [22, 29, 38, 42]. The energy of at≈le−ast (cid:126)ω = E0/2 39.5eV. For TDSE is solved in its full dimensionality including all 39.5eV < (cid:126)ω < 54.4eV, a−single≈photon does not inter-particle interactions. The laser field is linearly provide sufficient energy to ionize the He+ ion. Thus, polarized and treated in dipole approximation. The TPDI can only occur if the two electrons exchange interaction operator is implemented in both length and energy during the ionization process. In a temporal velocity gauge, such that gauge independence can be picture, this implies that the “first”, already ejected, explicitly checked. In the TDCC scheme the angular electron still has to be close to the nucleus when the part of the wave function is expanded in coupled secondphotonisabsorbed,i.e.,bothphotonshavetobe spherical harmonics. For the discretization of the absorbedquasi-simultaneously(ornonsequentially). For radial functions, we employ a finite element discrete photon energies larger than the ground state energy of variable representation (FEDVR) [43–46]. A local DVR the He+ ion ((cid:126)ω > 54.4eV), an independent-particle basis within each finite element leads to a diagonal picture is applicable for long pulses: each electron representation of all potential energy matrices. The absorbs one photon and electron-electron interaction is sparse structure of the kinetic energy matrices enables a priori not required for double ionization to occur. The efficient parallelization, giving us the possibility to firstelectronisreleasedfromtheHeatomwithanenergy employ pulses with comparably long durations (up to a of E1 = (cid:126)ω I1, while the second electron is released few femtoseconds) in our simulations. For the temporal from the He−+ ion with an energy of E2 = (cid:126)ω I2. − propagation of the wave function, we employ the short Here, I1 24.6eV (I2 54.4eV) is the first (second) ≈ ≈ iterative Lanczos (SIL) method [47–49] with adaptive ionization potential of helium. For shake-up satellites time-step control. thepartitioningofionizationpotentialsisdifferent(I(cid:48) = 2 Dynamical information is obtained by projecting I2/n2),andsoarethepeakpositionsE1(cid:48),2,buttheoverall the wave packet onto products of Coulomb continuum picture of sequential and independent photoionization states. As these independent-particle Coulomb wave events remains unchanged. functions are not solutions of the full Hamiltonian, For ultrashort pulses of a few hundred attoseconds, projection errors are, in principle, inevitable. However, the notion of sequentiality loses its meaning. The since we are able to propagate the wavepacket for long breakdown of the independent-particle picture and times after the conclusion of the pulse, errors in the strongcouplingbetweentheoutgoingelectronsisinthat asymptotic momentum distribution can be reduced to case not imposed by the necessity of energy-sharing but the one-percent level by delaying the time of projection is enforced by the ultrashort time correlation between untilthetwoelectronsaresufficientlyfarapartfromeach the two photoemission events occurring within Tp. other [23]. Electron-electron interaction therefore plays a decisive Most of the results presented were obtained at role in the correlated final momentum distribution. In a mean photon energy of (cid:126)ω = 70eV, which would particular,theelectronsarepreferablyemittedinaback- correspond to the sequential regime for long pulses. We to-back configuration at approximately equal energy choose the vector potential to be of the form A(t) = sharing, corresponding to a Wannier ridge configuration ˆzA sin2(πt/(2T ))sin(ωt) for 0 < t < 2T . The [11]. 0 p p duration T corresponds to the FWHM of the sine- A key indicator for sequential TPDI is that for p squared envelope function. The peak intensity was sufficiently low intensities (when ground state depletion cdheopsleetnioansaIn0d=th1r0e1e2-oWr/mcomre2ptohoetnosnuereffetchtastagreronuengdligsitbaltee. (cid:82)is∞ne(cid:82)g∞ligIib(tl)eI),(t(cid:48))tdhte(cid:48)dttotalT2y,iewldhesrcealTes iswitthhe dPusDerqaItio∝n −∞ t ∝ p p Inordertoreachconvergenceoftheangulardistribution, of the laser pulse [23, 37]. This is an immediate singleelectronangularmomentauptovaluesofl = consequence of two independent subsequent emission 1,max l = 10 were used. The highest total angular processes, the probability for each of which increases 2,max momentumincludedinthetimepropagationwasL = linearly with T , such that PDI (PI)2 T2. max p seq ∼ ∼ p 3. Forextractingthefinalprobabilitydistributions,only Equivalently,foreachofthetwoprocessesawell-defined the two-photon channels L = 0 and L = 2 were taken transitionrateW =lim PI/T exists. Thisimplies Tp→∞ p into account. The radial grid was composed of FEDVR thatthetotalratePDI/T ofthetwo-stepprocessgrows seq p elementsof4a.u.withorder11, withanextensionupto linearly with T in the limit of long pulses. By contrast, p 3 (a) 10−1 ¯hω−I1−E2 ¯hω−I2 ¯hω−I1 ¯hω−I2+E2 (b) 60 10 75as 8 300as V] 40 6 750as e [ u.] 10−2 1500as E2 20 4 b. 4500as 2 r [a 0 0 /Tp 10−3 0 20 40 60 ) (c) 60 10 1 E ( 8 DIP 10−4 eV] 40 6 [ 2 4 E 20 2 10−50 10 20 30 40 50 60 00 20 40 60 0 E1 [eV] E1 [eV] Figure 1. (a)Doubleionization(DI)ratePDI(E)/Tp (i.e.,DIprobabilitydividedbythepulseduration)forTPDIbyanXUVpulse at (cid:126)ω = 70eV with different pulse durations Tp. For sufficient pulse duration, the DI rate converges to a stable value except near the peaks of the sequential process. (b) and (c) show the two-electron energy spectrum PDI(E1,E2) for (b) Tp = 300as and (c) Tp=750as. thenonsequentialordirectdoubleionizationprobability the first electron is already far from the nucleus, the PDI scaleslinearlywithT andaconvergedtransition electrons cannot exchange a sufficient amount of energy. nonseq p rate exists in the limit W =lim PDI /T . For each final state, there is a maximum delay t(ii) Tp→∞ nonseq p c For ultrashort pulses, the scaling of the ionization between ionization events that can lead to that specific yield with T varies between T and T2 highlighting the energy sharing. This implies that the pulse has to be p p p non-uniform convergence over different regions of the considerably longer than this maximal delay in order to electron emission spectrum and the breakdown of the resolve all contributions to a specific final state. distinctionbetweendirectandindirectprocesses. Fig.1a In order to estimate the size of effect (ii), we illustrates the dependence of the energy differential employ a simple classical model: the first electron electron emission probability (projection of the joint is emitted with energy ESI = (cid:126)ω I1. In order − energy distribution Fig. 1b,c, onto the E1 (or E2) axis) to reach a specific final state with energies (E1,E2), for different pulse durations, divided by T , dW/dE = the liberated electron has to gain or lose the energy p PDI(E)/Tp. This quantity converges to a duration- ∆E = min(|ESI−E1|,|ESI−E2|) by interacting with independent cross section value (apart from constant the second electron. Therefore, the first electron can be factors) except in the regions near E = (cid:126)ω I1 and atmostadistancerSI(t(cii))=1/∆E fromthecoreatthe E = (cid:126)ω I , i.e., those values of the energy w−here the moment of the second photon absorption. This leads to 2 − sequential process is allowed [40]. The peak areas grow a critical time linearlywithTp indicativeofanoverallquadraticscaling 2(cid:112)α(α+1) ln(cid:16)2α+2(cid:112)α(α+1)+1(cid:17) characteristic for the sequential process (cf. Fig. 2a). If t(ii) = − , (1) one divides the yield contained in the peak areas by Tp2, c (2ESI)3/2 theresultisjustproportionaltotheproductofthesingle with α=E /∆E. SI ionization cross sections for one-photon absorption from Likewise, the spectral width of the pulse gives a theHegroundstateandone-photonabsorptionfromthe (i) corresponding time t = 1/∆E. Linear scaling should He+ ground state. c be observed for pulse durations T much longer than c The region within which the linear scaling prevails (i,ii) (i,ii) (i,ii) t . Setting T 10t leads to good agreement c c c is determined by the pulse duration for two different ≈ with the full numerical simulation (Fig. 2b). Moreover, reasons: both criteria give similar results thereby precluding a (i) Due to Fourier broadening, the photon energy is clear distinction between them. Fig. 2b displays the not well defined for a finite pulse, limiting the energy (i,ii) estimates T and the fraction of double ionization c resolution. Thus, if the broadened sequential peak probability that scales linear with T as a function of p overlaps with the final energy of interest, the long-pulse emission energy and pulse duration, limit PDI(E) T can not be observed. p ∝ PDI(E,T ) T (ii) There is an intrinsic maximum time delay PDI(E,T )= p max , (2) rel p PDI(E,T ) T between ionization events that can lead to a specific max p combination of final energies of the ejected electrons. where T = 4.5fs is the longest pulse we used. max When the second electron is ionized at a time when PDI takes on the value one when the double ionization rel 4 (a) 10 7 (b) 60 2 − totalTPDIyieldPDI PDI(E=Eeq) 50 10−8 PDI(E=41eV) 1.5 40 u.] 10−9 arb. eV] 30 1 [ [ d E el 10 10 yi − 20 0.5 10 11 − 10 10 12 0 0 − 75 125 250 500 1000 2000 4000 75 125 250 500 1000 2000 4000 Tp [as] Tp [as] Figure 2. (a)Scalingoftwo-photondoubleionizationyieldswithpulsedurationTp (FWHMofthesin2 XUVpulses)at(cid:126)ω=70eV. ThegreenpointsarethetotalionizationyieldPDI,theredsquaresgivethedifferentialyieldatequalenergysharingPDI(E=Eeq), with Eeq = (2(cid:126)ω+E0)/2, and the blue diamonds give the differential yield at E = 41eV. The dashed lines show fits to quadratic andlinearscalingwithTp forthetotalandsinglydifferentialyield. (b)ContourplotofPrDelI(E,Tp). Avalueof1forPrDelI (whitein thecolorscaleusedhere)markstheregionwherelinearscalingofthesinglydifferentialyieldwithpulsedurationTp isobserved. The orange lines indicate the positions of the peaks from the sequential process. The violet and green lines indicate the pulse durations (i) (ii) Tc andTc afterwhichlinearscalingoftheyieldwithTp isexpectedduetoFourierbroadeningofthesequentialpeakandbecause ofthemaximumtimedelaybetweenthephotonabsorptions(seetext). probability at energy E shows linear scaling with pulse (θ =0◦) and calculating the probability for the second 1 duration. We note that the estimate of effect (ii) could electron to be emitted into the forward half-space θ 2 ≤ be validated in a time-independent perturbation theory π/2 or backward half-space θ >π/2. The probabilities 2 calculation. ThelatterdoesnotshowFourierbroadening thus defined are butintroducesaneffectivecutofffortheinteractiontime t(cii) because of the limited box size. P±(E1,E2)= For long enough pulses, there is an additional (cid:90) interestingfeatureatenergiesE =(cid:126)ω I andE = 4π2 P(E1,E2,θ1=0◦,θ2)sinθ2dθ2, (3) (cid:126)ω I + , with = 2 2/1n2 the−exc1i−taEt2ion ene2rgy θθ22><ππ//22 2 2 n to t−he n-tEh excitedEstate −in He+. At these energies, wherethefactor4π2 stemsfromintegrationoverφ and 1 sequential ionization via the excited ionic (shake-up) φ . The forward-backward asymmetry is then given by 2 state nl is allowed. We discuss this is in more detail in | (cid:105) P+(E ,E ) P−(E ,E ) section 5. (E ,E )= 1 2 − 1 2 , (4) A 1 2 P+(E ,E )+P−(E ,E ) The non-uniform scaling with T described here 1 2 1 2 p shouldoccurforanyphotonenergywherethesequential which varies in the range [ 1,1]. Values close to zero process is allowed. This is confirmed by calculations at indicate vanishing correlat−ion between the electrons, (cid:126)ω = 91eV, shown in Fig. 6. At these higher photon while large absolute values identify strong angular energies, the ionized electrons obtain higher momenta, correlations. Positive values ( > 0) indicate a such that larger box sizes are required in the simulation preference for ejection of both eAlectrons in the same for the same pulse duration. Therefore, the maximum direction while negative values ( <0) indicate ejection durationofthelaserpulsewasrestrictedtoTp =1.5fs. in opposite directions. Note Athat (E1,E2) is not A symmetricunderexchangeofE andE ,astheemission 1 2 4. Angular correlations direction of the electron with energy E1 is fixed in the laser polarization direction. Analogously, the reduced Additional information on the dynamics of the two one-electron asymmetry (E ) can be determined by 1 A ionized electrons can be extracted from the angular integrating P±(E ,E ) over the energy of the second 1 2 correlations in the TPDI process. To that end, we electron,i.e.,P±(E )=(cid:82) P±(E ,E )dE ,and (E )= 1 1 2 2 1 A introducetheforward-backwardasymmetrydistribution (P+(E ) P−(E ))/(P+(E )+P−(E )). 1 1 1 1 (E ,E ), obtained by fixing the ejection direction of Fig. −3 shows the asymmetry of TPDI at (cid:126)ω = 1 2 A one electron in the direction of the laser polarization 70eV photon energy for pulses of different duration 5 1 ¯hω−I1−E2 ¯hω−I2 ¯hω−I1 ¯hω−I2+E2 75as 300as 750as 0.5 1500as 4500as ) 1 E 0 ( A 0.5 − 1 − 0 10 20 30 40 50 60 E1 [eV] Figure 3. Forward-backwardasymmetry (E1)forTPDIbyanXUVpulseat(cid:126)ω=70eV,fordifferentpulsedurationsTp. Thegray A linesshowtheexpectedpositionsofthepeaksforthesequentialprocess(withandwithoutshake-up). T , from T = 75as up to T = 4500as. For the onlyelectronsemittedinoppositedirectionareobserved p p p shortest pulses, the electrons are dominantly ejected in bothin“sequential”((cid:126)ω >54.4eV)and“nonsequential” opposite directions independent of energy, as observed (39.5eV < (cid:126)ω < 54.4eV) TPDI [23]. The main previously [11]. As the duration is increased, a differenceisthatinnonsequentialTPDI,onlythatregion stable pattern emerges: at the “sequential” peaks, is energetically accessible, such that no other angular the electrons are essentially uncorrelated, leading to configurations are observed. vanishing asymmetry. As most electrons are ejected in this channel, the total (energy-integrated) asymmetry 5. Shake-up interferences is very small for long pulses. However, for energies in between the two main peaks at E =(cid:126)ω I and E = (cid:126)ω−I2, theelectronsareemitted1inoppo−site1directio2ns. W(Ee re(cid:126)tωurnIn2o+w2t)oatnhdeloawdedrit(iEonal(cid:126)sωtrucIt1ures2)atenheirgghieesr This is precisely because these final state energies are ≈ − E ≈ − −E visible in Figs. 1 and 3. They correspond to shake- reachedonlywhenthetwoelectronsareejectedinsucha up satellites in He+ which can serve as intermediate configuration. This back-to-back Wannier-like emission states in sequential TPDI. In the shake-up process, near equal energy sharing remains pronounced even for the He+ ion is left in an excited state, while the free long pulses. electron obtains an energy of E(cid:48) = (cid:126)ω I 1 − 1 − En For energies outside the energy interval delimited (with the excitation energy to the n-th shell of n by the sequential peaks, the asymmetry is equally He+). EIn the long-pulse limit, this simply leads to the strong, but now positive pointing to the same emission appearanceofshake-upsatellitelinesatenergiesE(cid:48) and 1 direction for both electrons. When the second electron E(cid:48) =(cid:126)ω I + in the one-electron energy spectrum. 2 − 2 En is emitted in the same direction as the first one, the For ultrashort pulses, however, the nonsequential (or well-known post-collision interaction [50–53] tends to direct) double ionization channel becomes available as increase the asymmetric sharing of the available energy well and can lead to the same final states. Post- [11]. Thedividinglinebetweenthetwodifferentregimes collision interactions (PCI) lead to a broad distribution of ejection in opposite or in the same direction is quite of electron energies (see section 4), so that the electrons sharp and lies directly at the position of the sequential can obtain the same final energies of EPCI = E(cid:48) and 1 2 peaks. A more complete representation of the two- EPCI = E(cid:48) as the electrons emitted via He+(nl) in 2 1 electron energy and angular correlations is presented in the sequential process. Both indistinguishable pathways Fig. 4 for a pulse duration of T = 450as. While the lead to the same final state and thus to an interference p height gives the joint probability PDI(E ,E ), the color pattern in the double ionization yield, as observed 1 2 represents the asymmetry distribution (E ,E ). The in Fig. 1 and Fig. 3. This interference bears some 1 2 A borderline between positive and negative (i.e., 0, resemblance to the well-known exchange interference A A≈ white) is precisely near the peaks associated with the between e.g. photo-electrons and Auger electrons [54– sequential process. In the central region in between 57]. There is, however, a fundamental difference: while the “sequential” peaks the emission is preferentially on the exchange interference is intrinsically controlled by opposite sides while emission into the same hemisphere atomic parameters, namely the energy and lifetime prevails outside the main peaks. For completeness we (width) of the Auger electron, the novel interference note that in the region between the two main peaks, observedhereistrulyadynamicaleffectpresentonlyfor 6 E 2 E 1 Figure4. CombineddoubleionizationprobabilityPDI(E1,E2)andforward-backwardasymmetry (E1,E2)afterTPDIbyanXUV pulseat(cid:126)ω=70eVwithadurationof450as. Thez-axisgivesPDI(E1,E2)(inarbitraryunits),whileAthecolorencodestheasymmetry, withcyantobluesignifyingnegativevalues(ejectioninoppositedirection)andyellowtoredsignifyingpositivevalues(ejectioninthe samedirection). Vanishing correspondstowhite. ForenergieswherePDI(E1,E2)isnegligible,thecolorissettogray. A shortpulsesandcanbecontrolledbythepulseduration background contribution c is added to account for the bg T . different angular distributions of the different channels, p As the dependence of the yield on pulse duration which prevent complete interference. This gives is different for the different channels (proportional to PDI(E) (qΓ/2+E E )2 Tp for the nonsequential channel, proportional to Tp2 in PDI (E) ≈cbg+cF(E E )2+−(ΓF/2)2 , (5) the sequential channel), the observed spectrum strongly nonres − F changes with pulse duration. For short pulses (T < The simple fitting procedure used here only works well p 500as, cf. Fig. 1), the yield is completely dominated by for pulse durations T 1.5fs, as for shorter pulses, p ≥ the nonsequential channel without any trace of a shake- the employed approximation for the “nonresonant” up interference. As the pulse duration is increased, the backgroundbreaksdown,andtheshake-uppeakitselfis sequential channel with shake-up becomes increasingly lessstrongandconsiderablybroadened. Fig.5illustrates important. As expected from the interference of a the dependence of the obtained parameters on the pulse relatively sharp peak with a smooth background, the duration, confirming the expected behavior: for long peak resembles a Fano lineshape [58]. Thus, the pulses, the peaks converge to the satellite lines, i.e., position of the maximum is shifted from the position Lorentzians of vanishing width, such that E (cid:126)ω F expected in the limit of infinitely long pulses. Even I + (E (cid:126)ω I ), Γ 0, q→ −1. 2 n F 1 n E → − − E → | | (cid:29) for relatively long pulses (T = 4.5fs), similar to The overall strength I of the shake-up peak relative p F those produced in free electron lasers, the position of to the nonresonant background is obtained from the the shake-up peak in the one-electron energy spectrum integral over the Fano lineshape, I c (q2 1)Γ. F F ∝ − PDI(E) is shifted by a considerable fraction of an eV. This behaves approximately linear with T , confirming p The structural similarity to a Fano resonance (a quasi- the scaling of the sequential shake-up channel with T2 p discrete resonance due to the shake-up intermediate versus the scaling of the nonresonant background with state embedded in a smooth continuum due to the T (Fig. 5c). Also shown in Fig. 5a is the position E p max direct double ionization) suggests to characterize the of the maximum of the spectrum PDI(E) without any interference in terms of Fano resonance parameters further processing. for the position E (T ), width Γ(T ), and asymmetry Such effects could possibly be observed in FEL F p p parameter q(T ), as well as its strength I (T ) (Fig. 5). pulses, which reach focused intensities of up to p F p To apply Fano’s parametrization [58], the calculated 1016W/cm2. To confirm that the results shown energy spectrum PDI(E) is divided by the nonresonant here (calculated for 1012W/cm2) also apply for these spectrum PDI (E), taken to be proportional to the high intensities, we performed an additional calculation nonres singly differential cross section as predicted from the at a peak intensity of I = 5 1015W/cm2 with 0 · model by Horner et al [40, Eq. (8)]. Away from the a pulse duration of T = 4.5fs. The shape of p peaks, this fits the form of the spectrum very well. A the differential yield PDI(E) (not shown) is almost 7 58.5 2.6 6 E (a) (b) q (c) F 2.4 5 58 E I max 2.2 F q V] 57.5 V] 2 u.], 4 e e 1.8 b. 3 [ [ r E 57 Γ a 1.6 [ 2 F 56.5 ¯hω I2+ 2 1.4 I − E 1 1.2 56 1 0 1.5 2 2.5 3 3.5 4 4.5 1.5 2 2.5 3 3.5 4 4.5 1.5 2 2.5 3 3.5 4 4.5 T [fs] T [fs] T [fs] p p p Figure 5. Parameters of the shake-up interference peaks around 57eV for TPDI by an XUV pulse at (cid:126)ω = 70eV obtained from fittingtoaFanolineshape. (a)FanoresonanceenergyEF andpositionEmax ofthemaximuminthespectrum,(b)widthΓ,(c)Fano parameterq andintegratedyieldIF fromtheshake-uppathway. Seetextfordetails. unchanged compared to the result at 1012W/cm2 of this contribution, we became aware of work by peak intensity, even though the ground state survival Palacios et al [59] who also observe the interference probability is only 32%. The total double ionization between these different channels. probability is PDI = 36%, i.e., more than a third of the helium atoms in the laser focus are doubly ionized. 6. Summary Even though the yield in the shake-up peak is only 0.6% of the total yield for that duration, this could be We have presented a detailed study of the dynamics of seen in experiment as only the integrated one-electron the two-photon double ionization process in helium in energy spectrum has to be observed. Moreover, from theso-called“sequential”energyregimeforawiderange theposition,strengthandasymmetryoftheinterference of ultrashort pulse durations (75as to 4.5fs). We have peaks, information on the poorly known pulse duration shown how electron interaction and thereby correlation of FEL pulse “bursts” could possibly be deduced. enforced by the short pulse duration influences the The results shown up to now were obtained at a observed energy spectra and angular distributions. photon energy of (cid:126)ω = 70eV, where only the n = 2 The one-electron ionization rate PDI(E)/T con- p shake-up channel plays a role. While the qualitative verges to a stable value with increasing pulse duration behaviour of each shake-up peak is expected to be forenergiesawayfromthesequentialpeaks(E =(cid:126)ω I 1 independent of (cid:126)ω, new intermediate ionic states nl and E = (cid:126)ω I ), giving rise to a well-defined (dir−ect) become accessible at (cid:126)ω > I1 + n, converging| to(cid:105) differential d−oub2le ionization cross section. However, (cid:126)ω > E0 for n (where E0E 79eV is the near the peaks where the sequential process is allowed, − → ∞ ≈ − ground state energy of helium). This is demonstrated PDI(E)/T grows with T . We have thus observed a p p in Fig. 6 at a photon energy of (cid:126)ω = 91eV. As the non-uniform scaling of the double ionization probability shake-up probability strongly decreases with increasing with T . Even though in this spectral range the sequen- p n, only the peaks associated with n = 2 and n = 3 can tial process is allowed, both the direct and sequential clearly be identified at the pulse lengths used here (up co-exist, giving rise to interferences which are induced to Tp = 1.5fs). For longer pulses, more highly excited by the short time correlation between the two emission stateswouldstarttoplayaroleaswell. Inthatcase,one events. Thenonsequentialchannelwithoutshake-upand wouldneedtotakeintoaccountthatthepeaksforhigher the sequential shake-up channel, where the intermediate noverlapwitheachotheraswellaswiththenonresonant state after one-photon absorption is an excited state of background. theHe+ion,caninterfere. Inattosecondpulses,onlythe It should be noted that in order to observe these nonsequential channel contributes, while in long pulses interference effects, the asymptotic vectorial momenta (longer than the 4.5fs used here), the sequential shake- k ,k (i.e., not only the asymptotic energies (E ,E )) upchannelwoulddominate. Forpulsedurationsofafew 1 2 1 2 of the two pathways have to coincide. The shake- femtoseconds,asobtainedinfreeelectronlasers,thetwo up channel has an angular distribution considerably channels are similarly important, such that interference different from that of nonsequential channel, such that can be clearly observed. This interferences may open only partial interference between the final states is up the possibility to measure the duration of ultrashort expected. This leads to a rich structure in the observed XUV pulses in the femtosecond regime. angular distributions (not shown), a more detailed We have also found that the angular distributions analysis of which is in progress. During the preparation in the final states populated by nonsequential processes REFERENCES 8 10−2 ¯hω−I1−E2 ¯hω−I2 ¯hω−I1 ¯hω−I2+E2 450as n=4 n=3 n=3 n=4 750as 1500as u.] 10−3 b. r a [ /Tp 10−4 ) 1 E ( DI P 10−5 10−60 10 20 30 40 50 60 70 80 90 100 E1 [eV] Figure 6. Doubleionization(DI)ratePDI(E)/Tp (i.e.,DIprobabilitydividedbythepulseduration)forTPDIbyanXUVpulseat (cid:126)ω=91eVwithdifferentpulsedurationsTp. Shake-uppeaksupton=3arevisible. are strongly correlated. In ultrashort pulses, where the [9] Seres J, Yakovlev V S, Seres E, Streli C, TPDI process is necessarily nonsequential, the favored Wobrauschek P, Spielmann C and Krausz F 2007 emission channel is the Wannier ridge riding mode of Nat. Phys. 3 878 back-to-back emission at equal energies (cf. [11]). In [10] Zhang X, Lytle A L, Popmintchev T, Zhou X, longer pulses, back-to-back emission is strongly favored Kapteyn H C, Murnane M M and Cohen O 2007 in the region close to equal energy sharing, while for Nat. Phys. 3 270 strongly asymmetric energy sharing, the electrons are [11] Feist J, Nagele S, Pazourek R, Persson E, primarily emitted in the same direction. Schneider B I, Collins L A and Burgd¨orfer J 2008 arXiv.org:physics 0812.0373 Acknowledgments [12] ParkerJS,SmythESandTaylorKT1998J.Phys. B 31 L571 J.F., S.N., R.P., E.P., and J.B. acknowledge support by [13] Br¨auning H et al. 1998 J. Phys. B 31 5149 the FWF-Austria, Grant No. SFB016. Computational [14] Briggs J S and Schmidt V 2000 J. Phys. B 33 R1 time provided under Institutional Computing at Los Alamos. The Los Alamos National Laboratory is [15] Byron F W and Joachain C J 1967 Phys. Rev. 164 operated by Los Alamos National Security, LLC for the 1 National Nuclear Security Administration of the U.S. [16] Malegat L, Selles P and Kazansky A K 2000 Phys. Department of Energy under Contract No. DE-AC52- Rev. Lett. 85 4450 06NA25396. [17] Proulx D and Shakeshaft R 1993 Phys. Rev. A 48 R875 References [18] Lein M, Gross E K U and Engel V 2000 Phys. Rev. Lett. 85 4707 [1] Ackermann W et al. 2007 Nat. Photonics 1 336 [19] Rudenko A, de Jesus V L B, Ergler T, Zrost K, [2] Dromey B et al. 2006 Nat. Phys. 2 456 Feuerstein B, Schr¨oter C D, Moshammer R and Ullrich J 2007 Phys. Rev. Lett. 99 263003 [3] Goulielmakis E et al. 2008 Science 320 1614 [20] Staudte A et al. 2007 Phys. Rev. Lett. 99 263002 [4] Hentschel M, Kienberger R, Spielmann C, Reider GA,MilosevicN,BrabecT,CorkumP,Heinzmann [21] Antoine P, Foumouo E, Piraux B, Shimizu T, U, Drescher M and Krausz F 2001 Nature 414 509 Hasegawa H, Nabekawa Y and Midorikawa K 2008 Phys. Rev. A 78 023415 [5] Nabekawa Y, Hasegawa H, Takahashi E J and [22] Colgan J and Pindzola M S 2002 Phys. Rev. Lett. Midorikawa K 2005 Phys. Rev. Lett. 94 043001 88 173002 [6] Naumova N M, Nees J A, Sokolov I V, Hou B and [23] FeistJ,NageleS,PazourekR,PerssonE,Schneider Mourou G A 2004 Phys. Rev. Lett. 92 063902 B I, Collins L A and Burgd¨orfer J 2008 Phys. Rev. [7] NomuraYet al.2008Nat. Phys.advanced online A 77 043420 publication doi:10.1038/nphys1155 [24] Feng L and van der Hart H W 2003 J. Phys. B 36 [8] Sansone G et al. 2006 Science 314 443 L1 REFERENCES 9 [25] Foumouo E, Lagmago Kamta G, Edah G and [51] Gerber G, Morgenstern R and Niehaus A 1972 J. Piraux B 2006 Phys. Rev. A 74 063409 Phys. B 5 1396 [26] GuanX,BartschatKandSchneiderBI2008Phys. [52] RussekAandMehlhornW1986J. Phys. B 19911 Rev. A 77 043421 [53] Armen G B, Tulkki J, ˚Aberg T and Crasemann B [27] HasegawaH,TakahashiEJ,NabekawaY,Ishikawa 1987 Phys. Rev. A 36 5606 KLandMidorikawaK2005Phys.Rev.A71023407 [54] de Gouw J A, van Eck J, Peters A C, van der Weg [28] HornerDA,McCurdyCWandRescignoTN2008 J and Heideman H G M 1993 Phys. Rev. Lett. 71 Phys. Rev. A 78 043416 2875 [29] Hu S X, Colgan J and Collins L A 2005 J. Phys. B [55] Read F H 1977 J. Phys. B 10 L207 38 L35 [56] Rouvellou B, Rioual S, Avaldi L, Camilloni R, [30] Ivanov I A and Kheifets A S 2007 Phys. Rev. A 75 StefaniGandTurriG2003Phys.Rev.A67012706 033411 [57] V´egh L and Macek J H 1994 Phys. Rev. A 50 4031 [31] Nikolopoulos L A A and Lambropoulos P 2001 J. [58] Fano U 1961 Phys. Rev. 124 1866 Phys. B 34 545 [59] Palacios A, Rescigno T N and McCurdy C W [32] Nikolopoulos L A A and Lambropoulos P 2007 J. (unpublished) Phys. B 40 1347 [33] Pronin E A, Manakov N L, Marmo S I and Starace A F 2007 J. Phys. B 40 3115 [34] Sorokin A A, Wellhofer M, Bobashev S V, Tiedtke K and Richter M 2007 Phys. Rev. A 75 051402(R) [35] Barna I F, Wang J and Burgd¨orfer J 2006 Phys. Rev. A 73 023402 [36] Foumouo E, Antoine P, Bachau H and Piraux B 2008 New J. Phys. 10 025017 [37] IshikawaKLandMidorikawaK2005Phys. Rev. A 72 013407 [38] Laulan S and Bachau H 2003 Phys. Rev. A 68 013409 [39] Piraux B, Bauer J, Laulan S and Bachau H 2003 Eur. Phys. J. D 26 7 [40] HornerDA,MoralesF,RescignoTN,Mart´ınFand McCurdy C W 2007 Phys. Rev. A 76 030701(R) [41] Ullrich J, Moshammer R, Dorn A, D¨orner R, Schmidt L P H and Schmidt-B¨ocking H 2003 Rep. Prog. Phys. 66 1463 [42] Pindzola M S et al. 2007 J. Phys. B 40 R39 [43] McCurdyCW,HornerDAandRescignoTN2001 Phys. Rev. A 63 022711 [44] Rescigno T N and McCurdy C W 2000 Phys. Rev. A 62 032706 [45] Schneider B I and Collins L A 2005 J. Non-Cryst. Solids 351 1551 [46] Schneider B I, Collins L A and Hu S X 2006 Phys. Rev. E 73 036708 [47] Leforestier C et al. 1991 J. Comp. Phys. 94 59 [48] Park T J and Light J C 1986 J. Chem. Phys. 85 5870 [49] Smyth E S, Parker J S and Taylor K T 1998 Comput. Phys. Commun. 114 1 [50] Barker R B and Berry H W 1966 Phys. Rev. 151 14